3.137 \(\int \frac{1}{x \sqrt{3-3 x^2+x^4}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (2-x^2\right )}{2 \sqrt{x^4-3 x^2+3}}\right )}{2 \sqrt{3}} \]

[Out]

-ArcTanh[(Sqrt[3]*(2 - x^2))/(2*Sqrt[3 - 3*x^2 + x^4])]/(2*Sqrt[3])

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Rubi [A]  time = 0.0268767, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1114, 724, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (2-x^2\right )}{2 \sqrt{x^4-3 x^2+3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[3 - 3*x^2 + x^4]),x]

[Out]

-ArcTanh[(Sqrt[3]*(2 - x^2))/(2*Sqrt[3 - 3*x^2 + x^4])]/(2*Sqrt[3])

Rule 1114

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{3-3 x^2+x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{3-3 x+x^2}} \, dx,x,x^2\right )\\ &=-\operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{3 \left (2-x^2\right )}{\sqrt{3-3 x^2+x^4}}\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (2-x^2\right )}{2 \sqrt{3-3 x^2+x^4}}\right )}{2 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0091794, size = 40, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{6-3 x^2}{2 \sqrt{3} \sqrt{x^4-3 x^2+3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[3 - 3*x^2 + x^4]),x]

[Out]

-ArcTanh[(6 - 3*x^2)/(2*Sqrt[3]*Sqrt[3 - 3*x^2 + x^4])]/(2*Sqrt[3])

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Maple [A]  time = 0.003, size = 31, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{3}}{6}{\it Artanh} \left ({\frac{ \left ( -3\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}-3\,{x}^{2}+3}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(x^4-3*x^2+3)^(1/2),x)

[Out]

-1/6*3^(1/2)*arctanh(1/6*(-3*x^2+6)*3^(1/2)/(x^4-3*x^2+3)^(1/2))

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Maxima [A]  time = 1.6456, size = 27, normalized size = 0.68 \begin{align*} -\frac{1}{6} \, \sqrt{3} \operatorname{arsinh}\left (-\sqrt{3} + \frac{2 \, \sqrt{3}}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^4-3*x^2+3)^(1/2),x, algorithm="maxima")

[Out]

-1/6*sqrt(3)*arcsinh(-sqrt(3) + 2*sqrt(3)/x^2)

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Fricas [A]  time = 1.31135, size = 131, normalized size = 3.28 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (-\frac{3 \, x^{2} + 2 \, \sqrt{3}{\left (x^{2} - 2\right )} + 2 \, \sqrt{x^{4} - 3 \, x^{2} + 3}{\left (\sqrt{3} + 2\right )} - 6}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^4-3*x^2+3)^(1/2),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*log(-(3*x^2 + 2*sqrt(3)*(x^2 - 2) + 2*sqrt(x^4 - 3*x^2 + 3)*(sqrt(3) + 2) - 6)/x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{x^{4} - 3 x^{2} + 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x**4-3*x**2+3)**(1/2),x)

[Out]

Integral(1/(x*sqrt(x**4 - 3*x**2 + 3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{4} - 3 \, x^{2} + 3} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^4-3*x^2+3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^4 - 3*x^2 + 3)*x), x)